Question

Where does the graph of `y=(5x^4)-(x^5)` have an inflection point?

Answer 1

An inflection point is a point on the graph at which the concavity changes. To investigate concavity, we'll look at the sign of the second derivative:

`y=5x^4 - x^5`

`y'=20x^3 - 5x^4`

`y''=60x^2 -20 x^3 = 20x^2(3-x)`

Obviously `20x^2` is always positive, so the sign of `y''` is the same as the sign of `3-x`.
Which is positive for `x<3` and negative for `x>3`. At `x=3` the concavity changes.

An inflection point is a point on the graph, so we need:

when `x=3`, we get
`y = 5(3^4)-3^5 = 5(3^4)-3(3^4)=2(3^4) = 2(81)=162`

The point `(3, 162)` in the only inflection point for the graph of `y=5x^4 - x^5`